I'm a bit of what I call an "eHoarder". I have amassed a huge amount of digital media.

I have a media player, and a bunch of external hard drives that plug into it.

This seems like an archaic system that I would like to change. But I don't know what my options are, but it seems like there should be many better solutions.

I want:
To be able to use a remote to control the media (currently can)
To be able to play virtually any file type (currently can)
To be able to wirelessly add/sort/play media (current cannot)
A pretty and easy interface (on the screen, HUD?)

That's all I can think of at the moment.

If it helps or is important, we will be getting a nice new computer setup (system, monitor, keyboard, etc) so if a certain hardware component is needed we can make that adjustment fairly easily right now.

Personally I prefer to have a NAS (this can be a self-built home server) and a playback device (preferably mini-ITX or smaller). It's unclear to me what you intend by 'wirelessly add/sort/play' - if you want to connect your (permanently installed) devices wirelessly, I'd say: bad idea, there are far better solutions. If you mean: I want to play back from a laptop or a tablet, that's a different matter.

It all boils down to how you want to set up your playback environment. If you have some form of central storage (NAS/home server), you can set up playback however you want. I have a self-built home server myself (see sig) but I also have all my rooms wired with UTP cables.

For the wireless playback, there's Chromecast/AirPlay and similar olutions. Unless that's not what you want (then you need to be more specific )

If you explain what devices you'll be using people might be able to give you more specific tips.

eHoarder? And still having a lot of disk's around. That must be hard to manage.
Got a Thecus NAS with 5x3Tb drives in RAID5 so effective 11Tb of room. Much easier to manage and everything on the network so every device can access the data.
Kodi will access the shares from the NAS without any problem and I use my iPad and iPhone now as a remote device to register my evenings :-)

NAS - network attached storage. Basically a hard disk (or collection of them) which may be accessible to any machine on your network for reading and/or writing depending on privileges that can be granted to individual users. Basically it would replace your existing hard drives (if you get one with a suitably large capacity of course) and could then be accessed across your network.

Kodi set up on a suitable machine connected to your TV (or monitor on a "full" computer) can pull movies and TV shows stored on such a drive in a suitable format and play them to the TV/monitor. If you have a suitable TV type it may be that your TV's remote control can drive the Kodi screens and menus (called CEC control if you want to look it up), or if not you can buy media remotes which connect to the Kodi machine via a USB dongle or just use keyboard/mouse.

Your existing player should also be able to access a NAS drive that way too if you are comfortable with that and it works, at least from what the link you posted implied.